Do you really understand addition and subtraction?

This was the question Mark McCourt was getting us to ponder in the first session of Maths Teacher Network. He started off by running through the classic 1089 “trick” which I have written about before here.

These types of activity are such a powerful way to get students to carry out repeated practice to build fluency. If you handed out a sheet with 20 column subtractions and 20 column additions, you would get groans and do no favours for the general popularity of maths as a subject. If those repeated calculations have a purpose as they do here, the dynamic and energy in the classroom is completely different.

I’d used this before but had never given it the time that it deserves, nor had I extended it:

Try it with 4 digits, 5 digits, etc.

Go back to three digits and try it with a, b, c. In other words prove it algebraically.

Now try in a different base, e.g. Base 7!

I actually got a bit obsessed with this when I got home. I do love a good spreadsheet challenge, so I attempted to build something that would provide an algebraic proof of different numbers of digits and bases. Here is the result…

My reflection from this session was that even if I have “seen” something before, have I really investigated it deeply and could I use it more effectively with my classes? To which my answers were no and yes!

Another approach, back to 3 digits. The prime factorisation of 1089 is 9×11×11. Why is this? And what are the prime factorisations of the 4 digit magic numbers. Can these be explained?